Fundamentals of Plasma Physics

3.5 Drift equations 77

fields, the Lorentz equation becomes

m

d[vgc(t) +vL(t)]

dt

= q

[

E(xgc(t)) + (rL(t)·∇)E

]

+q[vgc(t) +vL(t)]×

[

B(xgc(t)) + (rL(t)·∇)B

]

.

(3.62)

The gyromotion (i.e., the fast cyclotron motion) is defined to be the solution of the equa-
tion

m

dvL(t)

dt

=qvL(t)×B(xgc(t)); (3.63)

subtracting this fast motion equation from Eq.(3.62) leaves

m

dvgc(t)

dt

= q

[

E(xgc(t)) + (rL(t)·∇)E

]

+q

{

vgc(t)×

[

B(xgc(t)) + (rL(t)·∇)B

]

+vL(t)×(rL(t)·∇)B

}

.

(3.64)

Let us now average Eq.(3.64) over one gyroperiod in which case termslinearin gyromotion
average to zero. What remains is an equation describing the slow quantities, namely

m

dvgc(t)

dt

=q

{

E(xgc(t))+vgc(t)×B(xgc(t)) +〈vL(t)×(rL(t)·∇)B〉

}

(3.65)

where〈〉means averaged over a cyclotron period. The guiding center velocity cannow be
decomposed into components perpendicular and parallel toB,

vgc(t) =v⊥gc(t) +v‖gc(t)B̂ (3.66)

so that

dvgc(t)

dt

=

dv⊥gc(t)

dt

+

d

(

v‖gc(t)B̂

)

dt

=

dv⊥gc(t)

dt

+

dv‖gc(t)

dt

B̂+v‖gc(t)d

B̂

dt

.(3.67)

Denoting the distance along the magnetic field bys, the derivative of the magnetic field
unit vector can be written, to lowest order, as

dB̂

dt

=

∂Bˆ

∂s

ds

dt

=v‖gcB̂·∇B,̂ (3.68)

so Eq.(3.65) becomes

m

[

dv⊥gc(t)

dt

+

dv‖gc(t)

dt

B̂+v^2

‖gc

B̂·∇B̂

]

= qE(xgc(t))
+qvgc(t)×B(xgc(t))
+q〈vL(t)×(rL(t)·∇)B〉.
(3.69)
The component of this equation alongBis

m

dv‖gc(t)

dt

=q

[

E‖(xgc(t)) +〈vL(t)×(rL(t)·∇)B〉‖

]

(3.70)